ANSWER
[tex]\frac{dy}{dx}\text{ = }\frac{-5}{(4x-5)^2}[/tex]
EXPLANATION
We want to differentiate y given:
[tex]y\text{ = }\frac{x}{4x\text{ - 5}}[/tex]
To differentiate fractions as this, we first spllit the numerator and denominator as:
U = x
V = 4x - 5
Now, differentiate them separately.
dU/dx = 1
dV/dx = 4
Now, use formula:
[tex]\frac{dy}{dx}\text{ =}\frac{V\frac{du}{dx}\text{ - U}\frac{dV}{dx}}{V^2}[/tex]
So, let us put those values in there:
[tex]\begin{gathered} \frac{dy}{dx}\text{ = }\frac{(4x\text{ - 5) }\cdot\text{ 1 - (x }\cdot\text{ 4)}}{(4x-5)^2} \\ \frac{dy}{dx}\text{ = }\frac{4x\text{ - 5 - 4x}}{(4x-5)^2} \\ \frac{dy}{dx}\text{ = }\frac{-5}{(4x-5)^2} \end{gathered}[/tex]
That is the answer
